Advanced Lectures in Mathematics Volume XIV Handbook of Geometric Analysis, No. 3

Editors: Lizhen Ji, Peter Li, Richard Schoen, and Leon Simon

International Press 高等教育出版社 www.intlpress.com HIGHER EDUCATION PRESS Advanced Lectures in Mathematics, Volume XIV Handbook of Geometric Analysis, No. 3

Volume Editors: Lizhen Ji, University of Michigan, Ann Arbor Peter Li, University of California at Irvine Richard Schoen, Stanford University Leon Simon, Stanford University

2010 Mathematics Subject Classification. 01-02, 53-06, 58-06.

Copyright © 2010 by International Press, Somerville, Massachusetts, U.S.A., and by Higher Education Press, Beijing, China.

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ISBN 978-1-57146-205-3

Printed in the United States of America.

15 14 13 12 2 3 4 5 6 7 8 9 ADVANCED LECTURES IN MATHEMATICS

Executive Editors Shing-Tung Yau Kefeng Liu Harvard University University of California at Los Angeles Zhejiang University Lizhen Ji Hangzhou, China University of Michigan, Ann Arbor

Editorial Board Chongqing Cheng Tatsien Li Nanjing University Fudan University Nanjing, China Shanghai, China

Zhong-Ci Shi Zhiying Wen Institute of Computational Mathematics Tsinghua University Chinese Academy of Sciences (CAS) Beijing, China Beijing, China Lo Yang Zhouping Xin Institute of Mathematics The Chinese University of Hong Kong Chinese Academy of Sciences (CAS) Hong Kong, China Beijing, China

Weiping Zhang Xiangyu Zhou Nankai University Institute of Mathematics Tianjin, China Chinese Academy of Sciences (CAS) Beijing, China Xiping Zhu Sun Yat-sen University Guangzhou, China

to Shing-Tung Yau in honor of his sixtieth birthday

Preface

The marriage of geometry and analysis, in particular non-linear differential equations, has been very fruitful. An early deep application of geometric analysis is the celebrated solution by Shing-Tung Yau of the Calabi conjecture in 1976. In fact, Yau together with many of his collaborators developed important techniques in geometric analysis in order to solve the Calabi conjecture. Besides solving many open problems in algebraic geometry such as the Severi conjecture, the characterization of complex projective varieties, and characterization of certain Shimura varieties, the Calabi-Yau manifolds also provide the basic building blocks in the superstring theory model of the universe. Geometric analysis has also been crucial in solving many outstanding problems in low dimensional topology, for example, the Smith conjecture, and the positive mass conjecture in general relativity. Geometric analysis has been intensively studied and highly developed since 1970s, and it is becoming an indispensable tool for understanding many parts of mathematics. Its success also brings with it the difficulty for the uninitiated to appreciate its breadth and depth. In order to introduce both beginners and non-experts to this fascinating subject, we have decided to edit this handbook of geometric analysis. Each article is written by a leading expert in the field and will serve as both an introduction to and a survey of the topics under discussion. The handbook of geometric analysis is divided into several parts, and this volume is the second part. Shing-Tung Yau has been crucial to many stages of the development of geo- metric analysis. Indeed, his work has played an important role in bringing the well-deserved global recognition by the whole mathematical sciences community to the field of geometric analysis. In view of this, we would like to dedicate this handbook of geometric analysis to Shing-Tung Yau on the occasion of his sixtieth birthday. Summarizing the main mathematical contributions of Yau will take many pages and is probably beyond the capability of the editors. Instead, we quote several award citations on the work of Yau. The citation of the Veblen Prize for Yau in 1981 says: “We have rarely had the opportunity to witness the spectacle of the work of one mathematician affecting, in a short span of years, the direction of whole areas of research.... Few mathemati- cians can match Yau’s achievements in depth, in impact, and in the diversity of methods and applications.”

Contents

A Survey of Einstein Metrics on 4-manifolds Michael T. Anderson ...... 1 1 Introduction...... 1 2 Brief review: 4-manifolds, complex surfaces and Einstein metrics . . . 2 3 ConstructionsofEinsteinmetricsI...... 5 4 ObstructionstoEinsteinmetrics...... 9 5 ModulispacesI...... 13 6 ModulispacesII...... 25 7 ConstructionsofEinsteinmetricsII...... 29 8 Concludingremarks...... 35 References...... 35

Sphere Theorems in Geometry Simon Brendle, Richard Schoen ...... 41 1 The Topological Sphere Theorem...... 41 2 Manifoldswithpositiveisotropiccurvature...... 42 3 The Differentiable Sphere Theorem...... 53 4 NewinvariantcurvatureconditionsfortheRicciflow...... 56 5 Rigidity results and the classification of weakly 1/4-pinched manifolds...... 63 6 Hamilton’s differential Harnack inequality for the Ricci flow ...... 67 7 Compactnessofpointwisepinchedmanifolds...... 68 References...... 72

Curvature Flows and CMC Hypersurfaces Claus Gerhardt...... 77 1 Introduction...... 77 2 Notationsandpreliminaryresults...... 77 3 Evolutionequationsforsomegeometricquantities...... 80 4 Essentialparabolicflowequations...... 85 5 Existenceresults...... 91 6 CurvatureflowsinRiemannianmanifolds...... 104 7 Foliationofa spacetimebyCMChypersurfaces...... 112 8 TheinversemeancurvatureflowinLorentzianspaces...... 123 vi Contents

References...... 125

Geometric Structures on Riemannian Manifolds Naichung Conan Leung ...... 129 1 Introduction...... 129 2 Topologyofmanifolds...... 131 2.1 Cohomology and geometry of differential forms...... 131 2.2 Hodgetheorem...... 134 2.3 Witten-Morsetheory...... 137 2.4 Vector bundles and gauge theory ...... 138 3 Riemanniangeometry...... 143 3.1 TorsionandLevi-Civitaconnections...... 143 3.2 Classification of Riemannian holonomy groups ...... 144 3.3 Riemanniancurvaturetensors...... 145 3.4 Flattori...... 146 3.5 Einsteinmetrics...... 149 3.6 Minimalsubmanifolds...... 149 3.7 Harmonicmaps...... 151 4 Orientedfourmanifolds...... 152 4.1 Gaugetheoryindimensionfour...... 153 4.2 Riemanniangeometryindimensionfour...... 155 5K¨ahlergeometry...... 156 5.1 K¨ahlergeometry— complexaspects...... 157 5.2 K¨ahlergeometry— Riemannianaspects...... 161 5.3 K¨ahlergeometry— symplecticaspects...... 165 5.4 Gromov-Wittentheory...... 168 6 Calabi-Yaugeometry...... 170 6.1 Calabi-Yaumanifolds...... 170 6.2 Special Lagrangian geometry...... 172 6.3 Mirrorsymmetry...... 174 6.4 K3surfaces...... 180 7 Calabi-Yau3-folds...... 183 7.1 ModuliofCYthreefolds...... 183 7.2 Curves and surfaces in Calabi-Yau threefolds...... 185 7.3 Donaldson-Thomas bundles over Calabi-Yau threefolds ...... 188 7.4 Special Lagrangian submanifolds in CY3 ...... 189 7.5 Mirror symmetry for Calabi-Yau threefolds...... 189 8 G2-geometry...... 190 8.1 G2-manifolds...... 190 8.2 Moduli of G2-manifolds...... 192 8.3 (Co-)associativegeometry...... 193 8.4 G2-Donaldson-Thomas bundles ...... 195 8.5 G2-dualities,trialitiesandM-theory...... 196 9 Geometryofvectorcrossproducts...... 197 9.1 VCPmanifolds...... 197 Contents vii

9.2 Instantonsandbranes...... 199 9.3 Symplectic geometry on higher dimensional knot spaces . . . . . 200 9.4 C-VCPgeometry...... 200 9.5 Hyperk¨ahler geometry on isotropic knot spaces of CY ...... 201 10 Geometryovernormeddivisionalgebras...... 203 10.1 Manifoldsovernormedalgebras...... 203 10.2 Gauge theory over (special) A-manifolds...... 205 10.3 A-submanifolds and (special) Lagrangian submanifolds . . . . . 205 11 Quaterniongeometry...... 207 11.1 Hyperk¨ahlergeometry...... 208 11.2 Quaternionic-K¨ahlergeometry...... 212 12 Conformalgeometry...... 212 13 GeometryofRiemanniansymmetricspaces...... 215 13.1 Riemanniansymmetricspaces...... 215 13.2 Jordanalgebrasandmagicsquare...... 217 13.3 Hermitianandquaternionicsymmetricspaces...... 219 14 Conclusions...... 221 References...... 222

Symplectic Calabi-Yau Surfaces Tian-Jun Li ...... 231 1 Introduction...... 231 2 Linearsymplecticgeometry...... 233 2.1 Symplecticvectorspaces...... 233 2.2 Compatiblecomplexstructures...... 235 2.3 Hermitianvectorspaces...... 238 2.4 4-dimensionalgeometry...... 241 3 Symplecticmanifolds...... 245 3.1 Almost symplectic and almost complex structures ...... 245 3.2 Cohomological invariants and space of symplectic structures...... 247 3.3 Moser stability and Darboux charts ...... 251 3.4 Submanifolds and their neighborhoods ...... 253 3.5 Constructions...... 254 4AlmostK¨ahlergeometry...... 259 4.1 Almost Hermitian manifolds, integrability and operators . . . . 259 4.2 Levi-Civitaconnection...... 263 4.3 Connections and curvature on Hermitian bundles ...... 266 4.4 ChernconnectionandHermitiancurvatures...... 271 4.5 Theself-dualoperator...... 275 4.6 Diracoperators...... 276 4.7 Weitzenb¨ock formulas and some almost K¨ahler identities . . . . 281 5 Seiberg-Wittentheory–threefacets...... 283 5.1 SWequations...... 284 5.2 Pin(2)symmetryfora spinreduction...... 289 viii Contents

5.3 The compactness and Hausdorff property of the moduli space...... 295 5.4 Genericsmoothnessofthemodulispace...... 298 5.5 Furuta’sfinitedim. Approximations...... 302 5.6 SWinvariants...... 311 5.7 Symplectic SW equations and Taubes’ nonvanishing...... 313 5.8 Symplectic SW solutions and Pseudo-holomorphic curves. . . . 319 5.9 Bordism SW invariants via finite dim. Approximations ...... 321 5.10 Mod2 vanishingandhomologytype...... 327 6 SymplecticCalabi-Yauequation...... 333 6.1 Uniquenessandopenness...... 334 6.2 A prioriestimates...... 335 7 SymplecticCalabi-Yausurfaces...... 337 7.1 Symplectic birational geometry and Kodaira dimension . . . . . 337 7.2 Examples...... 338 7.3 Homologicalclassification...... 344 7.4 Furtherquestions...... 348 References...... 352

Lectures on Stability and Constant Scalar Curvature D.H. Phong, Jacob Sturm ...... 357 1 Introduction...... 357 2 TheconjectureofYau...... 360 2.1 Constant scalar curvature metrics in a given K¨ahler class. . . . 360 2.2 The special case of K¨ahler-Einsteinmetrics...... 361 2.3 TheconjectureofYau...... 361 3 Theanalyticproblem...... 362 3.1 Fourth order non-linear PDE and Monge-Amp`ere equations...... 362 3.2 Geometricheatflows...... 363 3.3 Variational formulation and energy functionals ...... 363 4 The spaces Kk ofBergmanmetrics...... 365 4.1 Kodairaimbeddings...... 365 4.2 TheTian-Yau-Zelditchtheorem...... 366 5 The functional F 0 (φ)onK ...... 368 ω0 k 5.1 F 0 andbalanceimbeddings...... 369 ω0 5.2 F 0 and the Euler-Lagrange equation R − R =0...... 370 ω0 5.3 F 0 and Monge-Amp`eremasses...... 371 ω0 6 Notions of stability ...... 372 6.1 Stability in GIT ...... 372 6.2 Donaldson’s infinite-dimensional GIT ...... 381 6.3 Stability conditions on Diff(X) orbits...... 383 7 The necessity of stability ...... 385 7.1 The Moser-Trudinger inequality and analytic K-stability . . . . 385 7.2 Necessity of Chow-Mumford stability...... 387 Contents ix

7.3 Necessity of semi K-stability ...... 391 8 Sufficient conditions: the K¨ahler-Einsteincase...... 394 8.1 The α-invariant...... 395 8.2 Nadel’s multiplier ideal sheaves criterion ...... 395 8.3 The K¨ahler-Ricciflow...... 397 9 General L: energyfunctionalsandChowpoints...... 408 0 9.1 Fω andChowpoints...... 408 9.2 Kω andChowpoints...... 410 10 General L: theCalabienergyandtheCalabiflow...... 411 10.1 TheCalabiflow...... 411 10.2 Extremal metrics and stability ...... 412 11 General L: toricvarieties...... 414 11.1 Symplecticpotentials...... 415 11.2 K-stability on toric varieties ...... 415 11.3 The K-unstablecase...... 419 12 Geodesics in the space K of K¨ahlerpotentials...... 419 12.1 The Dirichlet problem for the complex Monge-Amp`ere equation...... 419 12.2 Method of elliptic regularization and a priori estimates . . . . . 420 12.3 Geodesics in K and geodesics in Kk ...... 423 References...... 427

Analytic Aspect of Hamilton’s Ricci Flow Xi-Ping Zhu ...... 437 Introduction...... 437 1 Short-timeexistenceanduniqueness...... 438 2 Curvatureestimates...... 441 2.1 Shi’slocalderivativeestimates...... 442 2.2 Preservingpositivecurvature...... 443 2.3 Hamilton-Iveypinchingestimate...... 444 2.4 Li-Yau-Hamiltoninequality...... 448 3 Singularitiesofsolutions...... 450 3.1 Canalltypesofsingularitiesbeformed...... 450 3.2 Singularitymodels...... 452 3.3 Canonicalneighborhoodstructure...... 456 4 Longtimebehaviors...... 457 4.1 TheRicciflowontwo-manifolds...... 458 4.2 TheRicciflowonthree-manifolds...... 461 4.3 Differential Sphere Theorems ...... 464 References...... 468